Operads
Operad theory is a field of abstract algebra concerned with prototypical algebras that model properties such as commutativity or anticommutativity as well as various amounts of associativity. Operads generalize the various associativity properties already observed in algebras and coalgebras such as Lie algebras or Poisson algebras by modeling computational trees within the algebra. Algebras are to operads as group representations are to groups. Originating from work in algebraic topology by Boardman and Vogt, and J. Peter May, it has more recently found many applications, drawing for example on work by Maxim Kontsevich and Thomas Willwacher on graph homology.
An operad can be seen as a set of operations, each one having a fixed finite number of inputs (arguments) and one output, which can be composed one with others; it is a category-theoretic analog of universal algebra.
The word "operad" was also created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer). Regarding its creation, he wrote: "The name 'operad' is a word that I coined myself, spending a week thinking of nothing else."
An operad without permutations (sometimes called a non-symmetric, non- or plain operad) consists of the following:
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